Understand and critique the assumptions underlying Markov multiple state models for long-term insurance benefits.

Disability income pays while the insured is sick and stops when she recovers or dies. The standard pricing tool is a Markov chain on three states, but every assumption it makes is testable and every one of them can fail.

A multi-state model uses a finite state space $\{0,1,\dots ,n\}$ with transition intensities

$${\mu }_x^{ij}=\underset{h\to 0^{+}}{\lim }\frac{{}_h{p}_x^{ij}}{h},i\ne j.$$

Premiums, reserves, and Thiele's equation all assume the chain is Markov. That single word packages several distinct claims about how lives behave.

Markov property. Given current state and current age, the future is independent of history. A life disabled 5 days and one disabled 5 years share the same recovery intensity ${\mu }_x^{10}$ at age $x$.

Intensities exist and are finite. Equivalently ${}_h{p}_x^{ij}=h{\mu }_x^{ij}+o(h)$ for small $h$.

At most one transition in $[x,x+h]$. Two or more jumps occur with probability $o(h)$. This is what lets you write Kolmogorov's forward equations.

Common mistakes

• Calling a duration-dependent model Markov. If ${\mu }^{10}$ depends on time since entering state 1, the chain is semi-Markov, not Markov.
• Assuming time-homogeneous when age matters. Setting ${\mu }_x^{ij}$ constant in $x$ forces mortality independent of age. Wrong for almost any life application.
• Forgetting probabilities sum to one. ${\sum }_j{}_t{p}_x^{ij}=1$. Candidates report ${}_t{p}_x^{00}$ as the survivor probability without adding ${}_t{p}_x^{01}$.

Bottom line

• Markov property: future transitions depend only on current state and age, not on duration in state or path taken.
• Transition intensities ${\mu }_x^{ij}$ exist, are finite, depend on age and current state only, and are the building block of multi-state pricing.
• States are exhaustive and mutually exclusive; the life is in exactly one at every instant, so ${\sum }_j{}_t{p}_x^{ij}=1$.
• At most one transition in an interval of length $h$; probability of two or more jumps is $o(h)$, which is what lets you write Thiele.

Exam shortcut

Read the stem for "depends on how long". If a transition depends on duration-in-state, Markov is violated and the default fix is to enlarge the state space. DECISION: Absorbing transitions only and Markov is fine. Reversible transitions and you must check duration dependence before trusting the reserve. "SMEAT" for the assumptions: States exhaustive, Memoryless given state, intensities Exist, At most one jump, Time-inhomogeneous in age only.

The full lesson (about 1,549 words, 10 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1c

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